Induced quasicocycles on groups with hyperbolically embedded subgroups

Let G be a group, H a hyperbolically embedded subgroup of G , V a normed G -module, U an H -invariant submodule of V . We propose a general construction which allows to extend 1-quasicocycles on H with values in U to 1-quasicocycles on G with values in V . As an application, we show that every group G with a nondegenerate hyperbolically embedded subgroup has dim H 2 b .G; `p.G// D 1 for p 1. This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undistorted with respect to the stable commutator length.